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A255595

Sylvester's sequence modulo 109.

2, 3, 7, 43, 63, 92, 89, 94, 23, 71, 66, 40, 35, 101, 73, 25, 56, 29, 50, 53, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32, 12, 24, 8, 57, 32

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OFFSET

0,1

COMMENTS

For most small primes, it's easy to see that they have no multiples in Sylvester's sequence (A000058) by considering the sequence modulo the prime in question. For example, Sylvester's sequence modulo 41 is 2, 3, 7, 2, 3, 7, 2, 3, 7, ...

But with 109, it isn't until A000058(25) modulo 109 that we encounter the repeated value of 32. From this point forward, the period {32, 12, 24, 8, 57} is infinitely repeated. The table in Sylvester (1880) is missing the 57.

REFERENCES

J. J. Sylvester, Postscript to Note on a Point in Vulgar Fractions. American Journal of Mathematics Vol. 3, No. 4 (Dec., 1880): 389, Table.

LINKS

Table of n, a(n) for n=0..75.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

FORMULA

a(0) = 2, a(n) = a(n - 1)^2 - a(n - 1) + 1 mod 109.

EXAMPLE

a(4) = 43 because a(3) = 7 and 7^2 - 7 + 1 = 43.

a(5) = 63 because 43^2 - 43 + 1 = 1807 = 63 mod 109.

MATHEMATICA

sylv109[0] := 2; sylv109[n_] := sylv109[n] = Mod[sylv109[n - 1](sylv109[n - 1] - 1) + 1, 109]; Table[sylv109[n], {n, 0, 108}]